Vice President Mike Pence during a rally with President Trump last week in Elkhart, Ind. While past vice presidents have played important roles maintaining the political coalitions of their ticket-mates, none wielded Mr. Pence’s independent influence over an administration’s political network and agenda.

Representative Jeb Hensarling of Texas needed a favor: Before retiring, he wanted to anoint a local activist as his successor. Mr. Hensarling, a veteran conservative, reached out to President Trump for help, but the White House hesitated to intervene, according to a person familiar with the overture.

Instead, Mr. Hensarling found a willing ally at Mr. Trump’s right hand: Vice President Mike Pence. Mr. Pence backed the congressman’s favorite, Bunni Pounds, last month in a tweet that blindsided key White House aides.

The eager assistance Mr. Pence provided a senior lawmaker reflected the outsize political portfolio that the vice president and his aides have seized for themselves as the 2018 elections approach. While Mr. Trump remains an overpowering personality in Republican politics, he is mostly uninterested in the mechanics of managing a political party. His team of advisers is riven with personal divisions, and the White House has not yet crafted a strategy for the midterms. So Mr. Trump’s supremely disciplined running mate has stepped into the void.

Republican officials now see Mr. Pence as seeking to exercise expansive control over a political party ostensibly helmed by Mr. Trump, tending to his own allies and interests even when the president’s instincts lean in another direction. Even as he laces his public remarks with praise for the president, Mr. Pence and his influential chief of staff, Nick Ayers, are unsettling a group of Mr. Trump’s fierce loyalists who fear they are forging a separate power base.

In addition to addressing dozens of party events in recent months, Mr. Pence has effectively made himself the frontman for America First Policies, an outside group set up to back Mr. Trump’s agenda. He has keynoted more than a dozen of its events this year, traveling under its banner to states including Iowa and New Hampshire. And Mr. Pence has worked insistently to shape Mr. Trump’s endorsements, prodding him in the contests for governor of Florida and speaker of the House, among others.

Word of the internal tensions is getting out beyond the walls of the White House: one prominent lawmaker said the complaints of high-ranking Trump officials were starting to circulate on Capitol Hill.

“They’re looking for people to stay on the team, not break away from the team,” Senator Lindsey Graham, Republican of South Carolina, said of the Trump side of the West Wing.

Mr. Trump and Mr. Pence remain on good terms personally, and the president has largely welcomed the vice president’s political guidance, according to people close to both men. And Mr. Pence has been intimately involved in planning for the 2020 campaign: He joined Mr. Trump for the meeting where the president told Brad Parscale, a digital strategist in the 2016 election, that he would manage the 2020 race. Mr. Pence stood behind Mr. Parscale, rubbing his shoulders, as Mr. Trump spoke.

Yet in at least two instances, the vice president, Mr. Ayers and other aides have badly overstepped. Mr. Pence recently abandoned an attempt to hire Jon Lerner, a Republican pollster close to Mr. Ayers, as a national security aide, after Mr. Trump discovered Mr. Lerner had helped lead attacks on him in the 2016 election.

Nick Ayers, the vice president’s chief of staff, speaking with Kellyanne Conway in the White House last year.

The quick dismissal of Mr. Lerner was widely seen as a brushback against Mr. Pence and Mr. Ayers, a way for Mr. Trump’s advisers to signal that they were closely watching the vice president’s office. Two senior White House officials said the Lerner episode made Mr. Trump more acutely aware of what these aides described as Mr. Pence’s empire-building.

Tensions also flared last year, after Mr. Ayers and another Pence aide were reported to have made suggestive comments to Republican donors about planning for an unpredictable 2020 election. Most brazenly, Marty Obst, a senior Pence adviser, told a Republican donor that Mr. Pence…

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